This work is focused on extending the well-studied concept of stochastic orders on univariate random variables to multivariate random variables. One method for accomplishing this extension is to impose univariate stochastic orders on scalarizations of random vectors. Adopting this approach, the thesis provides several equivalent characterizations of the stochastic order for random vectors. The dissertation studies the role of the multivariate stochastic-order relations, especially when incorporated as a constraint in stochastic optimization problems. A new two-stage stochastic optimization model, which incorporates a multivariate convex-order relation, is introduced and analyzed and its numerical solution is addressed. Optimality conditions are established for the single- and two-stage optimization problems with multivariate stochastic-order constraints. Several numerical methods are developed for solving the optimization problems with particular multivariate order constraints: stochastic dominance of second order and the multivariate counterpart of the increasing convex order. Several applications of the proposed models to finance and management problems are presented in detail with a discussion of numerical results.